Math Bridge: Charge, Energy, and Self-Discharge

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Math BridgeEnergy & PowerStruggle-friendly runway

How much of a ten-year current budget disappears inside the cell?

Separate charge from energy, compound shelf loss, and express the missing charge as an average current.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetPut self-discharge on the same current ledger as the product.
The chapter caseA 225 mAh CR2032, 3.0 V nominal, 2.0 V floor, and 1% loss/year.
What it buys youAn honest ten-year allowance before pulse and cutoff checks.

A field team faces an unresolved physical question: How much of a ten-year current budget disappears inside the cell? They must answer it before changing upper energy bound on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is upper energy bound. The middle card applies this page's relationship. The green card is lower energy bound. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Upper energy bound changes lower energy bound An input card leads through the page relationship to the lower energy bound result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Compounding is not a straight-line loss. The equivalent current spreads the accumulated loss across the same release interval used for the product budget.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for upper energy bound is 10.

  2. 2

    Name the relationship. Eupper=0.225x3.0=0.675 Wh; Elower=0.225x2.0=0.450 Wh retained=0.99¹⁰=90.44% Qlost=225x(1-0.9044)=21.51 mAh Ihidden=21.51 mAh/(10x8760 h)=0.246 uA

  3. 3

    Substitute the chapter fixture. Set upper energy bound to 10. The page ledger gives lower energy bound as 0.450 Wh.

  4. 4

    Read the result. Keep Wh beside the value. Use it only inside the technical boundary on this page.

Predict, then change upper energy bound

Try Predict the direction of lower energy bound. Move one control, calculate, then check your prediction.

10
Chapter baseline
Lower energy bound

Observe Compounding is not a straight-line loss. The equivalent current spreads the accumulated loss across the same release interval used for the product budget. Reset the control to 10 and compare lower energy bound.

Explain Only upper energy bound moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only upper energy bound moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only upper energy bound. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

Milliamp-hours count charge, while watt-hours also need voltage. A cell can lose charge internally while no load current appears in the product trace. Converting that loss into an equivalent average current makes it visible beside the circuit budget.

Battery Bruno: A current-equivalent shelf loss is an accounting tool, not a claim that a constant external current flowed.

2. Name every algebra move

1

Bound energyMultiply 0.225 Ah by 3.0 V and by the 2.0 V floor.

2

Compound retentionRaise 0.99 to the deployment years.

3

Find lost chargeMultiply capacity by one minus retention.

4

Find elapsed hoursMultiply years by 8,760.

5

Express the hidden averageDivide lost mAh by hours and convert mA to µA.

6

Compare with allowanceDivide by the chapter's 2.6 µA target.

3. Reproduce the chapter case

Eupper=0.225×3.0=0.675 Wh; Elower=0.225×2.0=0.450 Wh
retained=0.99¹⁰=90.44%
Qlost=225×(1−0.9044)=21.51 mAh
Ihidden=21.51 mAh/(10×8760 h)=0.246 µA

The hidden equivalent consumes about 9.4% of a 2.6 µA ten-year circuit allowance before pulse sag is checked.

4. Try one real input

TryChange deployment years and predict how compounded loss changes the hidden current.

Upper energy bound
Lower energy bound
Retained charge
Lost charge
Hidden average
Share of allowance

ObserveEnergy bounds do not change with deployment time. Retained charge falls, while the time-averaged equivalent changes gently because both lost charge and elapsed hours grow.

ExplainCompounding is not a straight-line loss. The equivalent current spreads the accumulated loss across the same release interval used for the product budget.

Technical boundaries.

This is a constant-rate shelf-loss model, not a cell qualification.

Rate
One percent per year is catalog-typical, not universal.
Voltage
The true delivered energy follows the discharge curve between the bounds.
Pulses
Internal resistance and reservoir-capacitor behaviour remain separate gates.

Correct, not complete: measure the selected cell across age, temperature, and pulse load.

5. Use the result in the design

Subtract shelf loss and margin before allocating circuit current, then verify cutoff and pulse delivery with the actual cell.

6. Record the evidence state

Keep chemistry, lot, capacity test rate, storage time and temperature, discharge curve, cutoff, pulse profile, and measured residual capacity.

7. Check yourself

Why is 225 mAh not an energy value?
Answer: It counts charge; energy also depends on delivered voltage.
Why compound 0.99 instead of subtracting exactly 10%?
Answer: Each year's percentage applies to the charge still retained.
Did 0.246 µA flow through the circuit?
Answer: No. It is the internal loss expressed on the circuit's average-current scale.
Honesty boundary.

The arithmetic uses the chapter's CR2032 capacity, voltage floor, and catalog-typical shelf rate.

Rate
One percent per year is catalog-typical, not universal.
Voltage
The true delivered energy follows the discharge curve between the bounds.
Pulses
Internal resistance and reservoir-capacitor behaviour remain separate gates.

Correct, not complete: measure the selected cell across age, temperature, and pulse load.

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